# Topology of hypersurface singularities

## TL;DR

This paper reviews the historical development and current understanding of the topology of hypersurface singularities, highlighting K"ahler's influential contributions and subsequent advances in the field.

## Contribution

It provides a comprehensive survey of the evolution of ideas on hypersurface singularities, emphasizing K"ahler's innovative methods and their impact.

## Key findings

- K"ahler introduced the 'square sphere' concept for singularity analysis.
- The survey covers developments in higher-dimensional hypersurface topology.
- Recent advances have expanded understanding of isolated hypersurface singularities.

## Abstract

K\"ahler's paper "\"Uber die Verzweigung einer algebraischen Funktion zweier Ver\"anderlichen in der Umgebung einer singul\"aren Stelle" offered a more perceptual view of the link of a complex plane curve singularity than that provided shortly before by Brauner. K\"ahler's innovation of using a "square sphere" became standard in the toolkit of later researchers on singularities. We describe his contribution and survey developments since then, including a brief discussion of the topology of isolated hypersurface singularities in higher dimension.